Problem 29 - College Algebra Essentials

Question

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x=4 y+1 \\ 3 y=1-4 x \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is x = 0.1 and y = -0.1

Step 1: Multiply the equations as needed to prepare for addition

The equations are already in the form AX+BY=C, but we want the coefficients of x to cancel each other out when the equations are added. To achieve this, multiply the first equation by 4 and the second by 3.\nEquations become:\n4*(3x)=4*(4y+1) => 12x=16y+4\n3*(3y)=3*(1-4x) => 9y=3-12x

Step 2: Add the equations

By adding the equations, we get:\n12x + 9y = 16y + 4 - 12x \nSimplifying, this becomes:\n24x = 7y + 4. Divide by 24 to solve for x: \nx = (7y + 4)/24

Step 3: Substitute the expression of x into one of the original equations to solve for y

Let's substitute x into the first original equation:\n3((7y+4)/24) = 4y + 1\nMultiplying through by 24 we get:\n7y+4 = 32y + 6\nSolving for y, we find y = -0.1 or -1/10.

Step 4: Substitute y=-0.1 into the expression for x to find the paired x value

Substituting y=-0.1 into the expression for x, we find x = (7(-0.1)+4)/24 = 0.1 or 1/10.

Key Concepts

Algebraic Methods

Algebra is a broad part of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. One of its key applications is solving systems of equations, which involves finding the values of variables that satisfy multiple equations simultaneously. There are several algebraic methods to solve these systems, including graphing, substitution, elimination (or addition method), and matrix methods.

The addition method, also known as the elimination method, is particularly useful when dealing with linear equations. To use this method, you manipulate the equations in a system to get the coefficients of one variable to be opposites. Then, you add the equations together to eliminate that variable, making it simpler to solve for the remaining variable. This technique is handy when the system does not lend itself easily to substitution or when the equations are already in a convenient form for elimination.

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the case of two variables, these systems can be represented graphically as lines on a coordinate plane. The solution to the system is the point(s) where the graphs of the equations intersect, meaning the values of the variables at this point make all the equations true simultaneously.

There are three possible outcomes when solving a system of linear equations: one unique solution (the lines intersect at a single point), no solution (the lines are parallel and never intersect), or infinitely many solutions (the lines are coincident, meaning they overlap entirely). Understanding these possibilities helps in determining the most suitable method for solving a given system and in interpreting the results.

Substitution Method

The substitution method is another algebraic technique used to solve systems of equations. This approach involves solving one of the equations for one variable in terms of the others and then substituting this expression into the other equation(s). This process transforms the system into a single equation with one variable, which can typically be solved more straightforwardly.

To effectively use the substitution method, it's often best to choose the equation and variable that makes the substitution process easiest — usually, this means picking the equation where the coefficient of the variable to solve for is 1 or −1. After solving for one variable, you substitute the solution back into one of the original equations to find the value of the other variable, giving you the ordered pair that represents the solution to the system.